A Bishop surface with a vanishing Bishop invariant by Huang X., Yin W.

By Huang X., Yin W.

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Extra resources for A Bishop surface with a vanishing Bishop invariant

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63). 62) holds for each α ≥ 0, we see that f (z, u) = f ∗ (z, u) in the formal sense. Hence, f (z, u) is also given by a convergent power series. 11 is finally complete. 2. 11 have the same content. 5. 16. Let (M, 0) be a real analytic elliptic Bishop surface with the Bishop invariant vanishing and the Moser invariant s < ∞ at 0. Then any element in aut0 (M) is a holomorphic automorphism of (M, 0). 17. Let M be defined by a real analytic function of the following form: ∞ w = zz + z s + z s + akl z k zl .

18). Let z = rσ(τ, r) with u = r 2 and r > 0 be the uniquely determined conformal map from the unit disk ∆ := {τ ∈ C : |τ| < 1} to D(u) with A Bishop surface with a vanishing Bishop invariant σ(0, r) = 0, στ (0, r) > 0. Here, as defined before, D(u) = {z ∈ C1 : zz + z s + z s + E(z, z) < u = r 2 }. Then φ(τ) = (rσ(τ, r), r 2 ) is a holomorphic disk attached to M. Similarly, let z = rσ ∗ (τ ∗ , r) with u = r 2 and r > 0 be the conformal map from the disk ∆ to D∗ (u) with σ ∗ (0, r) = 0, σ ∗ τ ∗ (0, r) > 0.

5. 16. Let (M, 0) be a real analytic elliptic Bishop surface with the Bishop invariant vanishing and the Moser invariant s < ∞ at 0. Then any element in aut0 (M) is a holomorphic automorphism of (M, 0). 17. Let M be defined by a real analytic function of the following form: ∞ w = zz + z s + z s + akl z k zl . k,l≥0; k+l>s k−l=0 mod s Then M is biholomorphically equivalent to its normal form w = zz + z s + zs . 11. 4 (d) and (e); for (z, w) → (eiθ z, w) is an automorphism of (M, 0) whenever eisθ = 1.

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